Question

check out the attachment​

Answer 1

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Value of p = 12

$\bold{\large{\underline{\underline{\sf{StEp\:by\:stEp\:explanation:}}}}}$

GiVeN :

• $\alpha$ and $\beta$ are the roots of the equation.

Equation :

• x² - 8x + p = 0
• a² + b² = 40

To FiNd :

• Value of p

SoLuTioN :

$\alpha$ and $\beta$ are the roots of the equation

Compare the given equation with the general quadratic form :

• ax² + bx + c = 0

1. a = 1
2. b = - 8
3. c = p

Sum of roots :

$\hookrightarrow$ $\alpha$ + $\beta$ = $\sf{\dfrac{-b}{a}}$

$\hookrightarrow$ $\alpha$ + $\beta$ = $\sf{\dfrac{-(-8)}{1}}$

$\hookrightarrow$ $\alpha$ + $\beta$ = $\sf{\dfrac{8}{1}}$

$\hookrightarrow$ $\alpha$ + $\beta$ = $\sf{8}$ ---> (1)

Product of roots :

$\hookrightarrow$ $\alpha\:\beta$ = $\sf{\dfrac{c}{a}}$

$\hookrightarrow$ $\alpha\:\beta$ = $\sf{\dfrac{p}{1}}$

$\hookrightarrow$ $\alpha\:\beta$ = $\sf{p}$ ---> (2)

$\sf{\underline{Squaring\:both\:sides\:of\:eq^n\:1}}$

$\hookrightarrow$ $\sf{\alpha^2\:+\:\beta^2\:=\:8^2}$

$\hookrightarrow$ $\sf{(\alpha\:+\:\beta)^2\:=\:8^2}$

$\hookrightarrow$ $\sf{\alpha^2\:+\:2\:\alpha\:\beta\:+\:\beta^2\:=\:64}$

$\hookrightarrow$ $\sf{\alpha^2\:+\:2\:p\:+\:\beta^2\:=\:64}$

$\sf{\because{\:\alpha\:\beta\:=\:p\:\:from\:eq^n\:2}}$

$\hookrightarrow$ $\sf{\alpha^2\:+\:\beta^2\:=\:64\:-\:2p}$

$\hookrightarrow$ $\sf{40\:=\:64\:-\:2p}$

$\sf{\because{\alpha^2\:+\:\beta^2\:=\:40\:(Given)}}$

$\hookrightarrow$ $\sf{40\:-\:64\:=\:-\:2p}$

$\hookrightarrow$ $\sf{-24\:=\:-2p}$

$\hookrightarrow$ $\sf{\dfrac{-24}{-2}\:=\:p}$

$\hookrightarrow$ $\sf{\dfrac{24}{2}\:=\:p}$

$\hookrightarrow$ $\sf{12\:=\:p}$

•°• Value of p is 12